Introduction

Traditional approaches to analysing housing indexes, such as hedonic or repeat-sales models, can introduce selection bias or violate underlying assumptions. Moreover, they fail to capture key regional, social, or macroeconomic factors that play a significant role in driving housing price movements.

In the recent research Sijp et al. (2025) propose a new approach that applies Principal Component Analysis (PCA) to regional SA4 level housing indexes, enabling the extraction of key drivers of price movements and offering a deeper understanding of the forces shaping housing markets.

The PCA results show that the first three principal components capture much of the behavior of local price indexes, as a linear combination of their time series explains most of the variance in the housing index. Nevertheless, the PCA-based linear model lacks robustness across different time windows, and its coefficient estimates must be derived directly from the PCA procedure.

Our exploratory research extends the PCA approach by addressing these limitations through a factor linear model, which uses time series data directly, with independent variables serving as proxies for the PCA-derived principal components. We model the factor proxies as time series using forecasting techniques to obtain their projected values. The forecasted factor series are subsequently incorporated into the linear model to generate projected housing price indexes, with time-varying coefficients included to capture evolving dynamics over time.

The resulting factor model can be applied for a variety of purposes: - To understand the key drivers of housing price movements at a national and regional level. - To generate forecasts of housing price indexes based on projected factor values. - To assess the impact of macroeconomic variables on housing prices through the regression coefficients.

Methodologies and Datasets

Methodlogies

PCA

Before introducing the factor model, Principal Component Analysis (PCA) was first applied to the original housing indexes. It is therefore useful to briefly revisit PCA and how it allowed us to identify specific factors. This work was originally conducted by Will Sijp, the supervisor of this project, whose detailed analysis can be found in the references. What follows is a short recap.

PCA is a statistical method that reduces complex datasets into a smaller number of uncorrelated components, each capturing a dominant source of variation. In our project, the PC1 explains 96% of the variance, while PC2 and PC3 account for 2.5% and 0.7%, respectively. Together, these three components capture nearly all the variability in the housing indexes.

The loadings from the PCA highlight which regions contribute most strongly to each component. For example, PC2, labelled the “mining” factor, shows high positive loadings in resource-dependent regions, while service-oriented regions display negative loadings. This pattern suggests that PC2 captures the influence of the mining sector on housing markets. By contrast, PC1, the “market” factor, has positive loadings across nearly all regions, reflecting the common national trend in housing prices.

Factor-Based Substitutes for Principal Components

After defining the factors associated with each principal component (PC), we introduce a linear regression that combines selected housing indexes to approximate the PCs, which are referred to as factor proxies. The objective is to identify interpretable city or regional combinations that closely mirror the PCA-derived components. A proxy is considered valid if it achieves a correlation of at least 96% with the corresponding PC, providing statistical evidence of its suitability. This procedure is not strictly systematic but rather heuristic, involving an iterative search for the most representative mix of cities or regions. More specifically:

  • For PC1 proxy the national spread (U), the factor model is specified as: \(\mu_r = \beta_r U + \varepsilon\)

  • For the PC2 proxy, the mining spread (Perth–Sydney spread, \(\delta_{PS}\)), the factor model is specified as: \(\mu_r = \beta_r U + \lambda_r \delta_{PS} + \varepsilon\)

  • For PC3 proxy the lifestyle spread (WIP)

Forecasting of Factors

Once the factor proxies were defined, we modelled each factor series separately using time series forecasting methods. The process included exploratory analysis, stationarity checks, and STL decomposition to characterise the data. Guided by these diagnostics, we selected suitable ARIMA-type models, enhancing them with Fourier terms to capture seasonality and intervention dummies to address structural breaks.

Integration into a Factor Modelling Framework

The linear model regresses each regional housing index on the factor proxies, yielding coefficients that represent the sensitivity of local markets to the underlying factors: \(\mu_r = \beta_{National} + \lambda_r \delta_{Mining} + \gamma_r \delta_{lifestyle} + \varepsilon\), the coefficients β, λ and γ are estimated for each region r. These coefficients help to capture the response of the local market to each factor. Model performance was evaluated using R-squared and residual diagnostics to check that key assumptions held. Forecasts of the factor proxies were then substituted into the model to obtain projected housing indexes. To account for changing dynamics, coefficients were re-estimated with rolling windows, allowing them to vary over time and better capture evolving market conditions.

Datasets

The datasets used in this project come primarily from the Australian Bureau of Statistics (ABS), covering housing price indexes and selected macroeconomic indicators. In addition, we make use of internally developed datasets, including preliminary outputs such as PCA results and estimated static coefficients. All datasets are organised and available in the data folder:

  • Housing indexes (log transformed) on both city level and SA4 level, named city_indexes and indexes_city_and_sa4
  • PCA results including PC scores and loadings, named df_pcs, df_eofs_city, and df_eofs_sa4
  • Factor proxies identified from PCA results, named df_factor_trends
  • Regression coefficients estimated from the factor proxies, named df_reg_coefs
  • Geographic data for SA4 regions, named SA4_2021_AUST_SHP_GDA2020.shp
  • Economic data including GDP, exchange rate, and mining expenditure, named data.xlsx, key_economic_data.xlsx, and mining_expenditure.xlsx (WIP)

Data Limitation

This project faces several dataset-related limitations, including but not limited to the following:

  • Historical Scope: PCA results and coefficients are based on 25 years of data (1998–2023), which may not fully reflect future patterns.
  • Macroeconomic Breadth: Available economic indicators are narrow and may miss other important drivers of housing prices.
  • Geographic Coverage: Indexes are only available at city and SA4 levels, limiting insights into finer sub-regional dynamics.
  • Measurement Bias: Indexes rely on hedonic and repeat-sales methods, which involve assumptions that may introduce bias.
  • Proxy Validity: Factor proxies are heuristic choices and may not perfectly represent the underlying principal components.
  • Model Assumptions: Linear regression assumes simple linear relationships, overlooking possible nonlinear interactions.
  • Data Sensitivity: Outcomes are highly data-driven and depend strongly on the chosen datasets and methods.
  • Structural Shocks: Events like the mining boom, COVID-19, or policy shifts may not be well captured by static models or PCA.
  • Endogeneity Risk: Factor spreads (e.g., mining, lifestyle) may themselves be influenced by national trends, complicating interpretation.
  • Forecast Stability: Forecasts rely on static coefficients, which may not hold under changing macroeconomic or demographic conditions.

Exploratory Analysis

The exploratory analysis utilises visualisations to characterise the dynamics of the national trend (U) and the Perth–Sydney spread (δPS), thereby clarifying their underlying properties.

Time-series plots:

  • U rises steadily from 2003 to 2012, transitions into a slower growth phase until 2019, then undergoes a sharp regime shift during 2020–2022 (the COVID period) followed by persistently elevated and volatile levels, with growth rates remaining time-varying throughout.

  • δPS displays a prolonged, uneven rise from 2003 to 2012, followed by a sharp regime shift, then COVID-related fluctuations, with the level adjusting and showing time-varying patterns.

Seasonal plots:

  • U has apparent within-year slopes being dominated by the strong upward trend; no systematic month-of-year effects.

  • δPS shows little evidence of month-to-month seasonality, with no particular month standing out, though its overall level varies between years.

Seasonal sub-series plot:

  • Each monthly panel of U shows a steady upward rise with mean value across months remaining similar, reinforcing that the series is driven more by trend than by seasonality.

  • The δPS seasonal sub-series plot reinforces the lack of a dominant seasonal pattern, as the monthly profiles are similar in shape and the corresponding means remain relatively flat and close to zero.

Forecasting with the Factor Model

With the factors now established, we proceed to fit time series forecasting models to each series individually. As a first step, we examine the series in more detail. STL decomposition plots allow us to closely inspect the trend, seasonality, and remainder components, which helps determine the most suitable forecasting models for the series’ characteristics.

Stationary tests (ADF and KPSS) indicate that both the national index (U) and the Perth–Sydney spread (PS) are non-stationary, so differencing is required. However, based on economic intuition that the PS series may be mean-reverting, we applied the Zivot–Andrews (ur.za) test. This confirmed that PS could be stationary with several structural break.

The dataset was partitioned into training and testing subsets, with one year of observations allocated to the testing set. Given that the data are recorded at a monthly frequency and span only 25 years, this allocation represents an optimal balance between ensuring sufficient data for model training while preserving an adequate sample for evaluation.

National Trend

The STL decomposition of the national trend U reveals a smooth and persistent upward trajectory in the trend component, while the seasonal component is of relatively minor magnitude, however the annual seasonality is strong and repeating. To stabilise variation in the seasonal component, a Box–Cox transformation was applied using the Guerrero method, which produced a λ value of 0.9. Given its proximity to one, the transformation had a negligible effect on the modelling outcome. The remainder component displays mostly small noise, with some bigger shocks.

The modelling strategy was guided by exploratory data analysis and the STL decomposition, which suggested a random walk with drift, ARIMA(U ∼ pdq(0,1,0)), as an appropriate baseline specification.

To refine the model, we subsequently evaluated neighbouring specifications with alternative autoregressive and moving average terms, using AICc values and residual diagnostics as selection criteria.

In the modelling process, we evaluated a range of specifications informed both by diagnostic checks and prior knowledge. Inspection of the residuals suggested some semi-annual variation, and Fourier terms with K = 2 were added to account for potential higher-frequency seasonality. We also investigated seasonal extensions with P = 1, alongside piecewise specifications with knots aligned to major economic events: the Global Financial Crisis (2008), the end of the mining investment boom (2012), and the onset of the COVID-19 pandemic (2020). These variations were assessed with respect to residual behaviour, statistical significance, and their ability to capture shocks apparent in the remainder component.

Among the fitted models, two candidates emerged as particularly strong. The first, an ARIMA(2,1,1) with Fourier terms (K = 2) and drift. A second competitive specification combined Fourier terms (K = 2) with intervention dummies (step and pulse) and an ARIMA(2,1,1) structure.

## # A tibble: 8 × 8
##   .model        sigma2 log_lik    AIC   AICc    BIC ar_roots   ma_roots  
##   <chr>          <dbl>   <dbl>  <dbl>  <dbl>  <dbl> <list>     <list>    
## 1 step      0.00000672   1338. -2650. -2649. -2602. <cpl [26]> <cpl [1]> 
## 2 arima211  0.00000700   1331. -2640. -2639. -2599. <cpl [26]> <cpl [1]> 
## 3 arima110  0.00000712   1327. -2637. -2636. -2604. <cpl [25]> <cpl [0]> 
## 4 arima111  0.00000715   1327. -2635. -2634. -2598. <cpl [25]> <cpl [1]> 
## 5 sarima211 0.00000785   1312. -2611. -2611. -2589. <cpl [14]> <cpl [1]> 
## 6 autoarima 0.00000774   1308. -2609. -2609. -2598. <cpl [24]> <cpl [0]> 
## 7 piece     0.00000791   1312. -2606. -2606. -2573. <cpl [1]>  <cpl [25]>
## 8 rw_drift  0.0000416    1066. -2127. -2127. -2116. <cpl [0]>  <cpl [12]>

Although both models fall short of strictly meeting the residual whiteness criterion, particularly since the degrees of freedom were not clearly defined in the test results, we use them here mainly for relative comparison of performance. Visual inspection of the residual plots suggests that the residuals are approximately normally distributed, with only a small share of autocorrelation coefficients exceeding the significance bounds in the ACF plot. As a general rule of thumb, if fewer than about 5% of the lags lie outside these bounds, the residuals can reasonably be treated as white noise.

## # A tibble: 8 × 3
##   .model    lb_stat lb_pvalue
##   <chr>       <dbl>     <dbl>
## 1 arima211     18.4   0.103  
## 2 step         20.4   0.0595 
## 3 sarima211    21.7   0.0411 
## 4 autoarima    23.0   0.0279 
## 5 arima110     24.7   0.0165 
## 6 arima111     24.8   0.0157 
## 7 piece        28.9   0.00411
## 8 rw_drift    885.    0

Mining Trend

The mining trend (PS) oscillates around zero and appears mean-reverting with no evidence ofpermanent trend. Seasonality is present but relatively stable across years, while longer multi-year swings suggest the need for more flexible seasonal modelling. The residuals still display structure, with a clear AR(1) cut-off rather than pure white noise.

The mining trend (PS) captures the dynamics of the resource boom, which we interpret as a one-off structural break rather than a permanent driver. Prices surged sharply during the 2000s mining boom, collapsed around 2012, and have remained stagnant since. If this raw factor is carried forward into forecasts, the boom–bust cycle dominates and produces unstable predictions with excessively wide intervals.

To address this, we treat the boom as a temporary structural shock. We introduce a dummy variable (set to 1 during the boom/decline period and 0 otherwise) when modelling PS, and then forecast only the residual stationary component. This approach effectively assumes that another boom of comparable scale is unlikely.

The boom and decline phases were identified using the breakpoints function from the strucchange package, which detected three structural breaks in the series. The segment containing the global peak was classified as the “boom” period, followed by the “decline” period. These periods were then encoded into dummy variables for use in the modelling process.

To model the series with the dummy variables, essentially we fit an ARIMA model to the residuals after accounting for the boom and decline periods. The final selected model was an ARIMA(2,0,1) with the boom and decline dummies included as regressors.

## Series: PS 
## Model: LM w/ ARIMA(2,0,1) errors 
## 
## Coefficients:
##          ar1      ar2      ma1     boom  decline
##       1.8741  -0.8763  -0.3764  -0.0081   0.0014
## s.e.  0.0389   0.0388   0.0829   0.0041   0.0041
## 
## sigma^2 estimated as 4.359e-05:  log likelihood=1061.23
## AIC=-2110.45   AICc=-2110.16   BIC=-2088.33

Forecast Performance

For the forecasting stage, we use the best-performing models (ARIMA(2,1,1) with Fourier terms (K = 2) for U and ARIMA(2,0,1) for PS) identified earlier to project values 10 years ahead (120 months). We generate forecasts with prediction intervals at both 80% and 95%. These intervals show the range in which future values are likely to fall, giving us a measure of uncertainty, an 80% interval means there is an 8-in-10 chance that the true value will lie within that range.

For evaluation, we set aside one year of data (2023) as a test set. Although a typical split is about 20% of the dataset, our series only spans 25 years, so using 20% would cut off too much valuable information for forecasting. A one-year test period strikes a better balance.

The forecasts indicate that the national trend is expected to rise, with an 80% prediction interval width of 0.6845, reflecting higher uncertainty. In contrast, the mining trend (PS) shows a flatter path, with an 80% interval width of 0.4538. This narrower range reflects greater stability, largely because we excluded the one-off boom and decline period, which would otherwise make the series more volatile.

The test set results show that the chosen model for U performs consistently with very low error (RMSE = 0.0076, negligible relative to the scale of U). For PS, the model incorporating boom/decline dummies produces out-of-sample forecasts that are on average about 7% off, which is acceptable given the series’ inherently difficult-to-predict behaviour (mining cycle is more volatile than the national market). While the ACF1 values indicate some remaining autocorrelation in the residuals, overall both models demonstrate solid performance.

## # A tibble: 2 × 6
##   series       .model   .type    RMSE     MAE  MAPE
##   <chr>        <chr>    <chr>   <dbl>   <dbl> <dbl>
## 1 U (national) arima211 Test  0.00756 0.00487 0.261
## 2 PS (mining)  arima201 Test  0.0116  0.00807 6.83

Factor Model Application

Goodness of Fit

We now turn to evaluating how well the factor model captures the variation in individual major city and SA4 indexes. R² measures the share of variation in each major city or SA4 housing index that our factor model explains.

For major cities and rest-of-state: Most city and regions have an R² value above 0.98. The extremes highlight clear spatial patterns:

  • Top cities and rest-of-state: Large east-coast capitals and their surrounding belts.

  • Bottom cities and rest-of-state: Smaller or resource-dependent markets.

Top 5 and Bottom 5 Major City and Area by R²
Top 5
Bottom 5
Rank Region R² Rank Region R²
1 GREATER MELBOURNE 0.9985 15 REST OF WA 0.9697
2 REST OF NSW 0.9985 14 GREATER PERTH 0.9735
3 REST OF QLD 0.9983 13 REST OF NT 0.9800
4 GREATER SYDNEY 0.9981 12 GREATER HOBART 0.9857
5 GREATER BRISBANE 0.9979 11 GREATER DARWIN 0.9860

For SA4 regions: The SA4 regions shares the same stories. R² is tightly concentrated near 1.0 with the majority of SA4s exceed 0.98, with only a small left tail (a few around 0.92 – 0.95). The top and bottom lists identify the extremes and show recurring patterns across those regions:

  • Top 10 regions: Concentrated in the large east-coast capitals and surrounding belts.
  • Bottom 10 regions : Dominated by Western Australia and Queensland’s mining exposed regions.
Top 10 and Bottom 10 SA4 Regions by R²
Top 10
Bottom 10
Rank Region R² Rank Region R²
1 MELBOURNE - SOUTH EAST 0.9982 87 WESTERN AUSTRALIA - OUTBACK (NORTH) 0.9184
2 MELBOURNE - OUTER EAST 0.9979 86 MANDURAH 0.9460
3 BRISBANE - WEST 0.9975 85 WESTERN AUSTRALIA - OUTBACK (SOUTH) 0.9601
4 BRISBANE - SOUTH 0.9974 84 BUNBURY 0.9606
5 BRISBANE - NORTH 0.9974 83 WESTERN AUSTRALIA - WHEAT BELT 0.9635
6 ADELAIDE - WEST 0.9972 82 MACKAY - ISAAC - WHITSUNDAY 0.9678
7 MELBOURNE - NORTH EAST 0.9972 81 PERTH - SOUTH EAST 0.9691
8 ADELAIDE - CENTRAL AND HILLS 0.9972 80 PERTH - NORTH EAST 0.9695
9 SYDNEY - BAULKHAM HILLS AND HAWKESBURY 0.9970 79 PERTH - SOUTH WEST 0.9732
10 WIDE BAY 0.9970 78 CENTRAL QUEENSLAND 0.9743

Both SA4 regions and major cities display highly concentrated R² distributions with major cities showing less variation.

The choropleth reinforces the findings from the summary table, as we can see that the east-coast capitals and coastal belts from Melbourne through Sydney into the South-East of Queensland show the strongest fit, with most regional NSW/VIC also high. While fit weakens inland and to the west, notably across Western Australia and along Queensland’s mining belts; The dark purple areas mark the lowest R² and are typically remote and resource-exposed regions with possible boom–bust timing and sparse transactions that our three factors don’t fully capture.

Hence, we’ll prioritise Western Australia and Queensland’s mining-exposed regions for residual diagnostics, where we expect autocorrelation and regime shifts for trial enhancements.

Residual Autocorrelation

At this stage, the model does not incorporate any time dynamics in the residuals. While it explains a substantial share of the variance for both cities and regions, the trending and autocorrelated nature of housing indexes means that the residuals themselves display extremely strong positive autocorrelation. This is evident in the Durbin–Watson statistics, which are close to zero across all regions: Major cities cluster around 0.02 to 0.05, while SA4 regions have a slightly broader range, extending up to 0.2, but still remain far below the benchmark value of 2. This pattern underscores the limitations of using a static regression framework without accounting for time-series dynamics.

## # A tibble: 2 × 6
##   region_level dw_min dw_max dw_mean dw_median  range
##   <chr>         <dbl>  <dbl>   <dbl>     <dbl>  <dbl>
## 1 major_city   0.0146  0.104  0.0338    0.0282 0.0891
## 2 sa4_name     0.0211  0.201  0.0680    0.0598 0.180

Coefficient Pattern

While R² and Durbin–Watson assess overall fit and residual behaviour, they do not show how regions load onto each factor. Examining coefficient patterns adds this context, revealing whether markets move uniformly with the national trend or diverge due to mining influences.

The distribution of market coefficients is tightly centred around 1.0 with relatively little variation. This indicates that most regions respond in a similar way to the national housing trend. In contrast, the mining coefficients are far more dispersed with a long right tail and several extreme outliers, suggesting that exposure to mining-related dynamics differs substantially across regions.

The scatterplot analysis reveals a clear negative relationship between the market and mining coefficients. Regions with a strong loading on the national market factor tend to exhibit weaker or even negative loadings on the mining factor and vice versa.

We tested ARIMAX to check if it could remove residual autocorrelation, which would indicate that static coefficients are adequate. ARIMAX keeps the regression coefficients fixed while applying an ARIMA model to the residuals. We initially fitted baseline ARIMA models (110, 011, 010, 111) and selected the best specification by balancing AICc values with the Ljung–Box test p-values, which indicate whether the residuals resemble white noise.

The ARIMA(1,1,0) model performed best, with 69.3% of SA4 regions showing residuals with p-values above 5%. For the remaining regions that failed the test, we applied two refinements: a seasonal model with seasonal (1,0,0) for regions with strong seasonality, and an AR(2) model for those with short-lag autocorrelation. These adjustments improved the overall pass rate to 86.4%.

Model Diagnostics: p-values and AICc across ARIMA Variants
pass mean_AIC model
0.6931818 -1923.782 Baseline ARIMA(1,1,0)
0.1000000 -1947.994 Short lag ARIMA(2,1,0)
0.0625000 -1873.096 Seasonal ARIMA(1,1,0)(1,0,0)[12]

Model Improvement with Time-dependent Coefficients

Forecasts for each major city and SA4 region are obtained by combining the projected factor trends with the static coefficients from the factor model. Specifically, the estimated loadings are applied to the forecasted values of factors to produce region-specific housing price index projections.

As the coefficients are fixed, the projected indexes evolve in parallel with the factor trends, scaled by their respective loadings, as illustrated in the accompanying figure. To capture greater regional heterogeneity, the subsequent stage of the analysis will extend the model to incorporate time-varying coefficients.

Rolling Ridge Regression

Rolling ridge regression is employed as small and noisy samples (particularly with volatile factors such as mining and lifestyle, can lead to instability in OLS-based rolling coefficient estimates) which adds a penalty term (λ) to the loss function to shrink coefficient estimates towards zero. This regularization helps reduce variance and improve out-of-sample performance, especially when predictors are correlated or the sample size is limited.

We set a 60-month window to captures regime shifts faster and used grid search to identify the global λ that minimises out-of-sample RMSE across all regions. This λ is then applied in a rolling regression framework to estimate time-varying regional coefficients. The optimal λ, 0.0475, indicates a light penalty that nonetheless helps stabilise the estimates. After that we fit rolling ridge regressions for each region using the chosen λ. The resulting time-varying coefficients capture evolving regional sensitivities to the national, mining and lifestyle trends.

We apply a rolling regression using a fixed 5-year window. For each region, the data are split into overlapping windows of equal length, each shifted forward by one month. Within each window we fit a penalised linear regression to estimate the intercept and factor coefficients. The resulting coefficients are recorded at the end of the window, then the window is advanced and the process repeats. This produces a time-series of coefficients for every region showing how the influence of the underlying factors evolves gradually over time.

  • To be added: more plots and analysis on rolling OLS result

—Progress here— # Boundedness Test on the Factor Trends (WIP)

(WIP - examine numbers since we switched to PS; Add more methodologies with the forecast model should be built at this stage)

To justify including δPS alongside the national trend U in the factor model, we need to show that δPS is more bounded over time, meaning it stays within a narrower range and has lower long-term volatility by comparing δPS and U using overall statistics and rolling-window measures.

  • The global summary (entire time period) of δPS and U covered several different measures, each of them captures different aspect of spread. The statistics includes:
    • Standard deviation: The average distance of each observation to the series mean. If δPS has a smaller standard deviation than U, it indicates that δPS is more stable and less volatile over the entire period.
    • Interquartile range: The spread of the medium 50% of the data, if δPS has a tighter interquartile range than U, then it is generally more stable.
    • Median aPSolute deviation: The median of abolute deviations from the mean. This measure is more robust to outliers. If the value of δPS is lower than U, it indicates that δPS has less extreme fluctuations.
    • Total range: The difference between the maximum and minimum values in the series, which examines the extreme swings in the series.

We next applied rolling windows of 12 and 24 months to the same statistics. Rather than using the entire time span at once, this approach slides a fixed-length window along the series, calculating the variability measures at each step using only the most recent 12 or 24 months of data. This method highlights how the volatility and dispersion of δPS and U evolve over time.

We use 12-month windows to smooth short-term fluctuations and capture typical annual housing cycles, helping assess if volatility is bounded within a year. The 24-month windows extend the view, showing whether this boundedness holds beyond yearly patterns and reflecting medium-term events.

To keep the report concise, we highlight two plots: the 12-month rolling standard deviation for short-term volatility and the 24-month rolling range for longer-term fluctuations.

Finally, we calculated the proportion of time (by 12 and 24 months respectively) in which δPS had a lower spread than U, giving a simple measure of how often it was more bounded.

The global ratios are all well below 1, meaning δPS has consistently lower spread than U across all four measures. This supports δPS being more bounded overall.

While the rolling-window plots and statistics give a more detailed picture:

  • The 12-month rolling standard deviation plot shows that δPS is often less volatile than U, but not consistently (occasional spikes narrow the gap).

  • The 24-month rolling range plot shows δPS has wider swings at times, but also periods where it is more stable than U, highlighting its tendency to avoid large swings over longer periods.

  • δPS has a lower spread than U in around 38.6% of the time for 12-month windows and 41.0% for 24-month windows, indicating it is more bounded (however less than half the time) in rolling windows.